Problem: Simplify and expand the following expression: $ \dfrac{3}{2p - 8}- \dfrac{1}{2p + 12}- \dfrac{5}{p^2 + 2p - 24} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $2$ out of denominator in the first term: $ \dfrac{3}{2p - 8} = \dfrac{3}{2(p - 4)}$ We can factor a $2$ out of denominator in the second term: $ \dfrac{1}{2p + 12} = \dfrac{1}{2(p + 6)}$ We can factor the quadratic in the third term: $ \dfrac{5}{p^2 + 2p - 24} = \dfrac{5}{(p - 4)(p + 6)}$ Now we have: $ \dfrac{3}{2(p - 4)}- \dfrac{1}{2(p + 6)}- \dfrac{5}{(p - 4)(p + 6)} $ The least common multiple of the denominators is: $ 4(p - 4)(p + 6)$ In order to get the first term over $4(p - 4)(p + 6)$ , multiply by $\dfrac{2(p + 6)}{2(p + 6)}$ $ \dfrac{3}{2(p - 4)} \times \dfrac{2(p + 6)}{2(p + 6)} = \dfrac{6(p + 6)}{4(p - 4)(p + 6)} $ In order to get the second term over $4(p - 4)(p + 6)$ , multiply by $\dfrac{2(p - 4)}{2(p - 4)}$ $ \dfrac{1}{2(p + 6)} \times \dfrac{2(p - 4)}{2(p - 4)} = \dfrac{2(p - 4)}{4(p - 4)(p + 6)} $ In order to get the third term over $4(p - 4)(p + 6)$ , multiply by $\dfrac{4}{4}$ $ \dfrac{5}{(p - 4)(p + 6)} \times \dfrac{4}{4} = \dfrac{20}{4(p - 4)(p + 6)} $ Now we have: $ \dfrac{6(p + 6)}{4(p - 4)(p + 6)} - \dfrac{2(p - 4)}{4(p - 4)(p + 6)} - \dfrac{20}{4(p - 4)(p + 6)} $ $ = \dfrac{ 6(p + 6) - 2(p - 4) - 20} {4(p - 4)(p + 6)} $ Expand: $ = \dfrac{6p + 36 - 2p + 8 - 20}{4p^2 + 8p - 96} $ $ = \dfrac{4p + 24}{4p^2 + 8p - 96}$ Simplify: $ = \dfrac{p + 6}{p^2 + 2p - 24}$